Considerations around Math Instruction for Students with Disabilities

Oct 30, 20124:30-6:30Specially Designed Instruction in Math PDU Session Two Text

Chapter Two: Learning to Calculate Outcomes for Session One Participants will review the foundational knowledge of the psychological processes of math through the development of number sense Participants will deepen their understanding between natural number sense and invented calculation tasks to solve problems that begins with multiplication Review the math psychological processes Perceptual subitizing Conceptual subitizing Magnitude of number Cardinality Counting Development of an Internal Number lineAddition and Subtraction Develop a scope and sequence sorting activity; sort the tasks as a group; review through this slide using a reveal. 4

The human brain has serious problems with calculations. Nothing in its evolution prepared it for the task of memorizing dozens of multiplication facts or for carrying out the multiple step operations required for two-digit subtraction. Our ability to approximate number quantities may be imbedded in our genes, but dealing with exact symbolic calculation can be in error prone ordeal.- Sousa

Example of CRA with Multiplication Ritas class is having a bake sale. Each student in Ritas class will bring 12 treats to sell. There are 25 students in Ritas class. How many treats will the class bring in altogether? 12 x 25Concrete Materials needed -Base Ten Blocks -Masking TapeConcrete Fill in the intersection of each row and column using a block that as the same dimensions Representation

Abstract 25 x 12

By age 4, children have created, refined, and selected algorithms for basic arithmetic (Griffin 2002)Every two years the childs number sense is reorganized based on experience to understand more complex tasks.This progression is predictable in about 80% of the population. 20% are slower and 20% are faster.4 year olds Operational Sense Global Quantity Schema Initial Counting Schema

more than less than 1 2 3 4 5Requires Subitizing Requires one-on-one Correspondence 6 year olds Operational Sense Internal Number line has been developed This developmental stage is a major turning point because children come to understand that mathematics is not just something that occurs out in the environment but can also occur inside their own heads. 1 10 20 30 40 50a little a lot 8 year olds Operational Sense Double internal number line has been loosley developed to allow for two digit operational problem solving Loosely coordinated number line is developed to allow for understanding of place value and solving double digit additional problems. 1 10 20 30 40 50a little a lot 1 10 20 30 40 50a little a lot 8 year old can now

10 year olds Operational Sense Double internal number line has been well developed to allow for two digit operational problem solving effeciently These two well developed number lines allow for the capability of doing two digit addition calculations mentally. 1 10 20 30 40 50a little a lot 1 10 20 30 40 50a little a lot A 10 year old can

Multiplication: Natural or Invented?

What is multiplication?

Ask the group to define and chart on a sheet of paper.19Imaging studies show that the brain recruits more neural networks during multiplication than during subtraction.

3+3= 93X3= 12Natural vs Invented + - x The big idea is that addition and subtraction are somewhat natural skills that are accused for survival, however multiplication is an invented tool and requires additional work to conquer the skill.Multiplication and Mistakes

The average adult makes multiplication mistakes about 10% of the time.Some multiplication facts such as 8x7 and 9x7 can take up to 2 seconds longer to solve and has an error rate of 25%.

Why? Three factors MemoryPattern and associations LanguageMemory Memory and Multiplication In a study in 1978 by Ashcroft they determined that memory plays a critical factor when doing calculations. It took less than a second to determine the results of 2+3 or 2x3, but about 1.3 second two solve 8+7 or 8x7. Why?

Because The accuracy of our mental representations of numerocity drops quickly with increasing number size.

We remember best what we learned first. When we begin learning our arithmetic fact, we started was simple problems containing small digits.

Because small digits appear more frequently than larger ones, we most likely receive much less practice with multiplication problems involving larger numbers.

Children in the primary grades encounter a sudden shift from their intuitive understanding of numerical quantities and counting strategies to the rote learning of arithmetic facts. Unfortunately, most children lose their intuition about arithmetic in the process. -SousaWhat is the tip on this bill?

20% tip on 76.10Did you memorizethe tip table? Memorization of multiplication facts; Is this intuitive? It still doesnt solve the problemNOOnes and Tens

are easiest because it aligns with our intuitive understanding of number; base ten system.Memorization of multiplication facts; Is this intuitive? It still doesnt solve the problemthis leaves 64 left to memorize

HOW?Memorization of multiplication facts; Is this intuitive? It still doesnt solve the problemThis cuts the number to only 32 to memorize.

This still doesnt solve the problem.So what does this mean for your teaching?

Turn and talk, come up with a few ideas and then chart as a group. 32Pattern Recognition and Associations Associative Memory Recital

When I show this word, what images pop up in your mind? 34Associative Memory three

Lets try it again with a number. What image pops up in your mind?35Associative Memory 6x7 ???One more time. What pops up in your mind? For most of you, there was probably no association made. We can make associations with numbers and can actually develop associations for addition, however multiplication is difficult because of how it is taught. 36Associations interfere with multiplication mastery Listen to a partner say these facts out loud.What kind of linguistic information do you hear? 6x9is547x8is568x8is64Rhythm? The brains strong pattern-seeking ability detects the rhythm of these entities when said aloud, thus making it difficult to keep these three expressions separate.

Carl Dennis lives on Allen Brian AvenueCarl Gary lives on Brian Allen AvenueGary Edwards lives on Carl Edward Avenue Who lives on Allen Brian Avenue?Where does Gary Edwards live?

Name Number Allen1Brian2Carol3Dennis4Edward5Frank6Gary7Lives On =3x4=12Carol Dennis lives on Allen Brian Avenue3x7=21Carol Gary lives on Brian Allen Avenue7x5= 35Gary Edwards lives on Carol Edward Avenue The brains ability to recognize patterns interferes with the childs ability to learn their multiplication facts. Learning multiplication actually interferes with understanding addition.

In 1998 a study by Miller discovered that students in third grade took more time to perform addition when they started learning the multiplication tables, and errors like 2+3= 6 began to occur.So what does this mean for your teaching?

Turn and talk, come up with a few ideas and then chart as a group. 41Language

Russian English Bilinguals 1 full second longer to solve a two digit problempresented in the second language Language and Multiplication 25 x 30=

Exact Approximate This does not mean that students who struggle with language processing will automatically struggle with mathematics, however language plays a critical role in learning multiplication.I want you to recite the alphabet and solve this two digit multiplication problem at the same time.

It is impossible because of where the brain stored multiplication facts in the same part of the brain used for language tasks like reciting the alphabet. We memorize facts using language.

This also explains why second language learns always revert to the first language to solve mathematical problems. It is less taxing on cognitive reserves. They have to take in the problem in the second language, translate to the first language, calculate the problem and then translate to the second language, thus taking a longer period of time.

When doing estimation, making approximations or determining the larger of two numbers the parietal lobes are activated on both right and left hemispheres. These are the number sense centers in the brain. They are naturally developed centers.

When doing exact calculations, the language processing areas in the frontal lobes are activated. As we became more sophisticated in our concept of number we needed to tie language to numbers. When we are problem solving we are using language. It is very difficult to do both calculations and language activates at the same time.

This doesnt mean that children with language processing issues have number issue as it is different parts of the brain. It just may mean they have difficulty expressing the numbers. 44There is strong evidence to suggest that the power of language and verbal memory can greatly enhance the childs mastery of multiplication; using poems and music are examples of this power of language.

So what does this mean for your teaching?

Turn and talk, come up with a few ideas and then chart as a group. 46Next time we meet 11-27-12Chapter 7 How the Brain Learns Mathematics; Assessing math learning disability; Best practice for remediation; CRA strategies